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hw4solutions_2

# hw4solutions_2 - Homework 4 Solutions Josh Hernandez 2.2...

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Homework 4 Solutions Josh Hernandez October 27, 2009 2.2 - The Matrix Representation of a Linear Transformation 2. Let β and γ be the standard ordered bases for R n and R m , respectively. For each linear transformation T : R n R m , compute [ T ] γ β b. T : R 2 R 3 defined by T ( a 1 , a 2 ) = (2 a 1 - a 2 , 3 a 1 + 4 a 2 , a 1 ). Solution: A transformation from a 2- to a 3- dimensional space has a 3 × 2 matrix: [ T ] γ β = 2 -1 3 4 1 0 . c. T : R 3 R defined by T ( a 1 , a 2 , a 3 ) = 2 a 1 + a 2 - 3 a 3 . Solution: [ T ] γ β = ( 2 1 -3 ) . 4. Define T : M × 2 × 2 ( R ) P 2 ( R ) by T ( a b c d ) = ( a + b ) + (2 d ) x + bx 2 . Let β = v 1 = 1 0 0 0 , v 2 = 0 1 0 0 , v 3 = 0 0 1 0 , v 4 = 0 0 0 1 , and γ = { 1 , x, x 2 } . Solution: T ( v 1 ) = 1, T ( v 2 ) = 1 + x 2 , T ( v 3 ) = 0, T ( v 4 ) = 2 x . Thus [ T ] γ β = 1 1 0 0 0 0 0 2 0 1 0 0 . 5. Let α v 1 = 1 0 0 0 , v 2 = 0 1 0 0 , v 3 = 0 0 1 0 , v 4 = 0 0 0 1 , and β = { 1 , x, x 2 } , and γ = { 1 } . c. Define T : M × 2 × 2 ( F ) F by T ( A ) = trace( A ). Compute [ T ] γ α Solution: T ( v 1 ) = 1, T ( v 2 ) = 0, T ( v 3 ) = 0, T ( v 4 ) = 1. Thus [ T ] γ α = ( 1 0 0 1 ) . 1

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d. Define T : P 2 ( R ) R by T ( f ( x )) = f (2). Compute [ T ] γ β . Solution: T (1) = 1 | x =2 = 1, T ( x ) = x | x =2 = 2, T ( x 2 ) = x 2 | x =2 = 4. Thus [ T ] γ β = ( 1 2 4 ) . 8. Let V be an n -dimensional vector space with an ordered basis β . Define T : V F n by the mapping T ( x ) = [ x ] β . Prove that T is linear. Solution: Denote β = { v 1 , . . . , v n } . Given v = c 1 v 1 + · · · + c n v n , and w = d 1 v 1 + · · · + d n v n in V , and some scalar a , T ( v + aw ) = T ( ( c 1 + ad 1 ) v 1 + · · · + ( c n + ad n ) v n ) = ( c 1 + ad 1 , . . . , c n + ad n ) = ( c 1 , . . . , c n ) + a ( d 1 , . . . , d n ) = T ( v ) + a T ( w
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